On Reduction of the Euler Equations by Means of Two-Dimensional Algebras

Abstract

A complete set of inequivalent two-dimensional subalgebras of the maximal Lie invariance algebra of the Euler equations is constructed. Using some of them, the Euler equations are reduced to systems of partial dierential equations in two independent variables which are integrated in quadratures. As well as developing approximate and numerical methods, finding exact solutions of the Euler equations (EEs) for an ideal incompressible fluid is an important problem of modern mathematical physics and hydrodynamics. There exist some ways to solve this problem. One of them is to use symmetry analysis [1, 2, 3]. We construct a complete set of inequivalent two-dimensional subalgebras of the maximal Lie invariance algebra of the EEs. Using some of them, we reduce the EEs to systems of partial dierential equations in two independent variables which can be integrated in quadratures. As a result, we obtain classes of exact solutions of the EEs that contain arbitrary functions. It is known [4], that the EEs ~t + (~ · ~